How can I make a "Movie Clip" node start at a certain frame of the scene?
Replace the "movie clip" node for an "Input Image" node and use Movie as source. 

Frames will give you control on the source's duration (in frames): How many frames do you want to use? 

Start Frame is the frame in the scene at which the video clip will start playing.

Offset allows you to select the first frame of the video source.



Cyclic Will make the cycle of frames start over, if the number of frames is different that those on the scene or if you are using any offset. If you want to hold a frame, before or after the specified number of frames or freeze a frame you would need to disable this box. 

Auto Refresh if disabled you will not see the sequence of images change.